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Topics referred to by the same term
Trace formula may refer to: Arthur–Selberg trace formula, also known as invariant trace formula, Jacquet's relative trace formula, simple trace formula
Trace_formula
Mathematical theorem
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group
Selberg_trace_formula
In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over
Arthur–Selberg_trace_formula
On the character of the representation of a reductive algebraic group
In mathematics, the local trace formula (Arthur 1991) is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation
Local_trace_formula
Expresses the number of points of a variety over a finite field
geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism
Grothendieck_trace_formula
{\displaystyle \pi _{1}(G)} . This is a (conjectural) version of the Lefschetz trace formula for Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} when X
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
Mapping theorem in topology
is a formula that counts the fixed points of a continuous mapping from a compact topological space X {\displaystyle X} to itself by means of traces of the
Lefschetz_fixed-point_theorem
Formula in analytic number theory
number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula. The Kuznetsov or relative trace formula connects Kloosterman sums
Kuznetsov_trace_formula
Mathematical conjecture about the Riemann zeta function
formulated a trace formula that is actually equivalent to the Riemann hypothesis. This strengthened the analogy with the Selberg trace formula to the point
Hilbert–Pólya_conjecture
Equation in Fourier analysis
trace" or "semiclassical trace" formula relates geometric and spectral properties of the underlying topological space. The spectral side is the trace
Poisson_summation_formula
Particular kind of exponential sum
H\subset G} be a subgroup. While the usual trace formula studies the harmonic analysis on G, the relative trace formula is a tool for studying the harmonic analysis
Kloosterman_sum
In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite
Behrend's_trace_formula
Extension of Lidskii's theorem
Banach spaces. The theorem should not be confused with the Grothendieck trace formula from algebraic geometry. Given a Banach space ( B , ‖ ⋅ ‖ ) {\displaystyle
Grothendieck_trace_theorem
In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a
Petersson_trace_formula
Sum of elements on the main diagonal
respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). The trace of an n × n square matrix A is
Trace_(linear_algebra)
Mathematical conjecture about elliptic curves
value of ap, the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication, trace ≠ 0) their formula states that the
Sato–Tate_conjecture
a total of 890 pages. 1983 – Selberg trace formula. Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with a total length
List of long mathematical proofs
List_of_long_mathematical_proofs
Theorem in linear algebra
(b)\right)} where tr {\displaystyle \operatorname {tr} } is the matrix trace and diag ( a ) {\displaystyle \operatorname {diag} (a)} is the diagonal
Schur_product_theorem
Theorem in abstract algebra
conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory
density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula (named after Martin Gutzwiller):
Quantum_chaos
Canadian mathematician (born 1944)
and 2002. Arthur is known for the Arthur–Selberg trace formula, generalizing the Selberg trace formula from the rank-one case (due to Selberg himself)
James_Arthur_(mathematician)
Conjecture on zeros of the zeta function
geodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved
Riemann_hypothesis
Type of graph in mathematics and physics
following trace formula for the Laplace operator on a graph when the transition amplitudes are independent of k {\displaystyle k} . The trace formula links
Quantum_graph
German mathematician
function). He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient
Kai_Behrend
Mathematical group
introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula. Roughly speaking, an endoscopic group H of G is a quasi-split group
Endoscopic_group
Mathematical concept
Meyer (2005), who derived the explicit formula of Weil via harmonic analysis on adelic spaces. Selberg trace formula Selberg zeta function The original prime
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
s_{0}/2} .[citation needed] Selberg trace formula Fischer, Jürgen (1987), An Approach to the Selberg Trace Formula via the Selberg Zeta-Function, Lecture
Selberg_zeta_function
Swiss-American physicist (1925–2014)
mechanics in chaotic systems. In that context, he developed the Gutzwiller trace formula, the main result of periodic orbit theory, which gives a recipe for
Martin_Gutzwiller
Norwegian mathematician (1917–2007)
theory into number theory, culminating in his development of the Selberg trace formula, the most famous and influential of his results. In its simplest form
Atle_Selberg
section 16) in their book Automorphic Forms on GL(2) using the Selberg trace formula. It was one of the first examples of the Langlands philosophy that maps
Jacquet–Langlands correspondence
Jacquet–Langlands_correspondence
Canadian mathematician
related to those for quaternion algebras. This book applied the adelic trace formula for G L ( 2 ) {\displaystyle \mathrm {GL} (2)} and quaternion algebras
Robert_Langlands
Korean educator (born 1978)
introduction to the stable trace formula". In Clozel, L.; Harris, M.; Labesse, J.-P.; Ngô, B. C. (eds.). The stable trace formula, Shimura varieties, and
Sug_Woo_Shin
forayed into a new territory in the field and created the relative trace formula in representation theory, an important tool in modern number theory
Hervé_Jacquet
Linear operators with a common spectrum
Urakawa, C. Gordon). In particular Vignéras (1980), based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric
Isospectral
Complex-valued function
\|X\|_{1}=\sum _{i}|\lambda _{i}(X)|} is the trace-class norm. One definition uses the exponential trace formula. For finite-dimensional matrices, we have
Fredholm_determinant
German mathematician (born 1958)
established a dynamical Lefschetz trace formula: it relates the trace of an operator on harmonic forms the local traces appearing at the closed orbits (on
Christopher_Deninger
Duality between the process of restricting and inducting in representation theory
the subject of group representations. See Selberg trace formula and the Arthur-Selberg trace formula for generalizations to discrete cofinite subgroups
Frobenius_reciprocity
Condition for a mathematical function to map some value to itself
fixed-point theorem Topological degree theory Tychonoff fixed-point theorem Trace formula Brown, R. F., ed. (1988). Fixed Point Theory and Its Applications. American
Fixed-point_theorem
Representation theory
denominator of the Weyl character formula vanish to high order at the identity element, so it is necessary to take a limit of the trace of an element tending to
Weyl_character_formula
Phenomenon in quantum systems
According to the trace formula, a quantum spectrum is not a result of a trace over all the positions, but it is determined by a trace over all the periodic
Quantum_scar
Australian mathematician
reductive groups", On the stabilization of the trace formula (PDF), Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications
Diana_Shelstad
American mathematician
Hejhal works on analytic number theory, automorphic forms, the Selberg trace formula and quantum chaos. From 1972 to 1974 he was a Sloan Fellow. In 1986
Dennis_Hejhal
Mathematical concept
local fields. Arthur’s work, which was motivated by the Arthur–Selberg trace formula, suggests a framework for understanding complex relationships in these
Arthur's_conjectures
Grothendieck–Teichmüller theory Grothendieck-Witt ring Grothendieck trace formula Grothendieck trace theorem Grothendieck pretopology Grothendieck topoi Grothendieck
List of things named after Alexander Grothendieck
List_of_things_named_after_Alexander_Grothendieck
Type of group in group theory
initiated by Robert Langlands. One of the main tool used there is the trace formula originating in Selberg's work and developed in the most general setting
Arithmetic_group
Conjectures connecting number theory and geometry
Harish-Chandra's approach on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. What was new in Langlands' work, besides technical
Langlands_program
Indian theoretical physicist
nuclear three body problem of triton. He developed an exact semiclassical trace formula to calculate level densities of spherical nuclei with no adjustable
Sudhir_Ranjan_Jain
James; Ellwood, David; Kottwitz, Robert (eds.). Harmonic Analysis, The Trace Formula, and Shimura Varieties. Vol. 4. Princeton: Clay Mathematics Institute
Grand_Riemann_hypothesis
Open-wheeled single-seater motorsport
best known are Formula One, Formula E, Formula Two, Formula Three, regional Formula Three and Formula Four. Common usage of "formula racing" encompasses
Formula_racing
class of X. Behrend's trace formula Behrend's trace formula generalizes Grothendieck's trace formula; both formulas compute the trace of the Frobenius on
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
American mathematician
Flicker is the author of a number of books including: Arthur's Invariant Trace Formula and Comparison of Inner Forms (2016) Drinfeld Moduli Schemes and Automorphic
Yuval_Flicker
Russian-American mathematician
new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with
Edward_Frenkel
Riemann hypothesis. Selberg class S Explicit formulae for L-functions Trace formula A directory of all known zeta functions This set index article includes
List_of_zeta_functions
Israeli mathematician
automorphic forms, L-functions, representation theory, and the Selberg–Arthur trace formula. In 1989 Lapid received from Tel Aviv University a B.Sc. and an M.Sc
Erez_Lapid
Parameterization of a rotation into a unit vector and angle
\omega } , this consistency can be ensured by generalizing the above trace formula as follows: θ = a t a n 2 ( − T r ( K R ) , T r ( R ) − 1 ) , {\displaystyle
Axis–angle_representation
French mathematician
Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands. [Drinfelʹd shtukas, Arthur-Selberg trace formula and Langlands correspondence]
Laurent_Lafforgue
Coca-Cola Company's recipe for Coca-Cola syrup
The formula for Coca-Cola syrup, which bottlers combine with carbonated water to create the famous soft drink, is a closely guarded trade secret that
Coca-Cola_formula
Type of generalization of periodic functions in Euclidean space
years around 1960, in creating such a theory. The theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory.
Automorphic_form
Algebraic geometry analog of a principal bundle in algebraic topology
Easy". math.ucr.edu. Retrieved 2022-11-22. Behrend, K. The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. PhD dissertation. Behrend
Torsor_(algebraic_geometry)
Australian mathematician (born 1981)
which he completed in 2002, producing the thesis "Limiting forms of the trace formula". He was supported by the Hackett Fellowship for postgraduate study
Akshay_Venkatesh
Mathematical object studied in the field of algebraic geometry
Compactifications and cohomology of modular varieties. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 551–582
Algebraic_variety
Russian mathematician
Kalinin and Ludvig Faddeev: A nonarithmetic derivation of the Selberg trace formula, Journal of Soviet Mathematics, vol. 8, 1977, pp. 171–199 Spectral theory
Alexei_Venkov
Belgian mathematician
of the Langlands philosophy. Deligne's conjecture on the Lefschetz trace formula (now called Fujiwara's theorem for equivariant correspondences). Brumer–Stark
Pierre_Deligne
Manufactured food designed for feeding infants
Infant formula, also called baby formula, simply formula (American English), formula milk, baby milk, or infant milk (British English), is a manufactured
Infant_formula
One of six awards by the Wolf Foundation
combinatorics. 2015 James G. Arthur Canada for his monumental work on the trace formula and his fundamental contributions to the theory of automorphic representations
Wolf_Prize_in_Mathematics
Expression in differential equations
of the above differential equation. Since the trace of A(x) is zero for all x ∈ I, Liouville's formula implies that the determinant is actually a constant
Liouville's_formula
MR 2108211. Some other references are Behrend, Kai (1991). The Lefschetz trace formula for the moduli stack of principal bundles (PDF) (Thesis). University
Quotient_stack
Romanian American mathematician
the Selberg trace formula". Journal of Functional Analysis. 53 (2): 151–201. doi:10.1016/0022-1236(83)90050-2. (See Selberg trace formula.) Connes, Alain;
Henri_Moscovici
conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selberg trace formula Artin conjecture Sato–Tate conjecture Langlands program modularity theorem
List_of_number_theory_topics
Generalization of matrix trace
categorical trace methods to prove an algebro-geometric version of the Atiyah–Bott fixed point formula, an extension of the Lefschetz fixed point formula. Ponto
Categorical_trace
Concept in class field theory
ISBN 978-3-540-06558-6, MR 0349635 Kottwitz, Robert (1984), "Stable trace formula: cuspidal tempered terms", Duke Mathematical Journal, 51 (3): 611–650
Weil_group
American mathematician
the University of California, San Diego. Her dissertation, Selberg's Trace Formula and Units in Higher Degree Number Fields, concerned number theory and
Dorothy_Wallace
Formula for the derivative of a matrix determinant
which is the Jacobi formula for matrices A with distinct nonzero eigenvalues. The following is a useful relation connecting the trace to the determinant
Jacobi's_formula
formulae of the general theory.) It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2 dim X det
Local_zeta_function
French mathematician (born 1946)
fourth powers is 19. With Henryk Iwaniec, he improved the Kuznetsov trace formula. In 1997, with Effinger and Herman te Riele, he proved the ternary Goldbach
Jean-Marc_Deshouillers
American mathematician (born 1942)
zeta functions of graphs, arithmetical quantum chaos, and the Selberg trace formula. Terras was elected a Fellow of the American Association for the Advancement
Audrey_Terras
theorem Theorem of the three geodesics Curve-shortening flow Selberg trace formula Selberg zeta function Zoll surface Besse, A.: "Manifolds all of whose
Closed_geodesic
Gases apart from nitrogen, oxygen, and argon in Earth's atmosphere
Trace gases are gases that are present in small amounts within an environment such as a planet's atmosphere. Trace gases in Earth's atmosphere are gases
Trace_gas
Soviet nuclear physicist
extended Gutzwiller's semiclassical theory of shell structure (Gutzwiller trace formula) to a quasiclassical theory of the nuclear shell structure (1977). This
Vilen_Strutinsky
Branch of mathematics that studies abstract algebraic structures
modular forms. Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann–Roch theorem
Representation_theory
Complex exponential in terms of sine and cosine
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric
Euler's_formula
Mathematical concept
Arthur, David Ellwood, and Robert Kottwitz (ed) Harmonic Analysis, the Trace Formula and Shimura Varieties, Clay Mathematics Proceedings, vol 4, AMS, 2005
Shimura_variety
Conjecture in algebraic geometry
to be completed in a second volume using the Grothendieck-Lefschetz trace formula and the Ran space. Ono (1965) used the Weil conjecture to calculate
Weil's conjecture on Tamagawa numbers
Weil's_conjecture_on_Tamagawa_numbers
Italian mathematician
arXiv:1607.03012 [math.AG]. Toën, Bertrand; Vezzosi, Gabriele (2017). "Trace formula for dg-categories and Bloch's conductor conjecture I". arXiv:1710.05902
Gabriele_Vezzosi
French-owned Formula One racing team
involvement of the sportscar manufacturer Automobiles Alpine in Formula One can be traced back to 1968, when the Alpine A350 Grand Prix car was built, powered
Alpine_F1_Team
Israeli mathematician
Character Sheaves and Some Generalization of the Lefschetz-Verdier Trace Formula) under supervision of Joseph Bernstein. From 1997 to 1999 he was a C
Alexander_Braverman
Moduli spaces of ramified covers
(approached with combinatorial means), and to use the Grothendieck trace formula and Deligne's estimations of eigenvalues of Frobenius (as explained
Hurwitz_space
Summatory function of the Möbius function
derivatives with respect to t are evaluated at t = 0. There is also a trace formula involving a sum over the Möbius function and zeros of the Riemann zeta
Mertens_function
French mathematician
With C. Moeglin, he wrote two large volumes establishing the stable trace formula for twisted spaces. Some recent publications are available on its website
Jean-Loup_Waldspurger
1970), Wolf Prize medallist, mathematician known for Arthur-Selberg trace formula and Arthur conjectures Jeffrey Brock (B.A. 1992), Dean of the School
List of Yale University people
List_of_Yale_University_people
Mathematical object
doi:10.4153/CMB-2002-049-1. MR 1941222. Kottwitz, Robert (1984). "Stable trace formula: cuspidal tempered terms". Duke Mathematical Journal. 51 (3): 611–650
Langlands_group
( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} Selberg's trace formula shows that for an hyperbolic surface of finite volume it is equivalent
Arithmetic_Fuchsian_group
Mathematical object
case of spherical pair. For example, the relative trace formula is obtained from the trace formula by this procedure. A few common examples of Gelfand
Gelfand_pair
Major type of automorphic form in mathematics
of level one automorphic forms for split classical groups using the trace formula, arXiv:1406.4247, Bibcode:2014arXiv1406.4247T Tsuyumine, Shigeaki (1986)
Siegel_modular_form
leading mathematician in representation theory and creator of the General Trace Formula Assikinack, Francis (1848) – Ojibwe historian and treaty negotiator
List of Upper Canada College alumni
List_of_Upper_Canada_College_alumni
Unsolved problem in mathematics
James; Ellwood, D.; Kottwitz, Robert E. (eds.). Harmonic analysis, the trace formula, and Shimura varieties: proceedings of the Clay Mathematics Institute
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Press 1987 Thomason, R.W.: Lefschetz–Riemann–Roch theorem and coherent trace formula. Invent. Math. 85, 515–543 (1986) Thomason, R.W., Trobaugh, T.: Higher
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
Completes the Langlands program for general linear groups over algebraic function fields
Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands." (Drinfeld shtukas, Arthur-Selberg trace formula and Langlands correspondence)
Lafforgue's_theorem
Formula in Lie theory
Eichler. A consequence of the Baker–Campbell–Hausdorff formula is the following result about the trace: tr log ( e X e Y ) = tr X + tr Y . {\displaystyle
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Mathematical function
In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.
Field_trace
TRACE FORMULA
TRACE FORMULA
Surname or Lastname
English
English : nickname from Middle English, Old French grace ‘charm’, ‘pleasantness’ (Latin gratia).English : from the female personal name Grace, which was popular in the Middle Ages. This seems in the first instance to have been from a Germanic element grīs ‘gray’ (see Grice 1), but was soon associated by folk etymology with the Latin word meaning ‘charm’.
Girl/Female
English
from Thracia.
Surname or Lastname
English (Kent)
English (Kent) : perhaps a variant of Treece.Altered spelling of German Treis, a topographic name for someone who lived by or owned an uncultivated piece of land used as pasture, from Middle Low German drīsch ‘fallow land’, or a habitational name from a place named with this word (in Hessian dialect treis), in Hesse or on the Mosel river. Alternatively, in some instances it may be from a short form of the personal name Andreas (see Andrew).
Girl/Female
American, Arabic, Australian, British, Chinese, Christian, Danish, English, French, German, Gujarati, Indian, Irish, Jamaican, Latin, Muslim, Portuguese, Swedish
Mercy; God's Favor; Grace; Grace of God; Kindness; Thanks; Love; Favour; Blessing; Charm; Good will
Surname or Lastname
English
English : probably from Middle English, Old French brace ‘arm’, also denoting a piece of armor covering the arm. In most cases it is probably a metonymic occupational name for a maker or seller of armor, specifically armor designed to protect the upper arms, but it could also have been a nickname for someone with strong arms (compare Armstrong) or a deformed or otherwise noticeable arm.
Male
English
Short form of English unisex Tracy, TRACE means "place of Thracius."
Girl/Female
English American
from Thracia.
Female
English
Feminine variant spelling of English unisex Tracy, TRACIE means "place of Thracius."
Surname or Lastname
English
English : perhaps a variant of Treece.
Boy/Male
Anglo Saxon American Latin Greek English French
Brave.
Girl/Female
Greek American French
Reaper; from Therasia.
Male
English
Variant spelling of English unisex Tracy, TRACEY means "place of Thracius."
Boy/Male
American, Anglo, Australian, British, Chinese, English, French
Fighter; Brave
Girl/Female
Latin American English Irish
Grace.
Boy/Male
Anglo Saxon American Greek
Brave.
Female
English
Feminine variant spelling of English unisex Tracy, TRACI means "place of Thracius."
Male
English
English surname transferred to unisex forename use, from a Norman baronial name TRACY means "place of Thracius."
Female
English
Feminine variant spelling of English unisex Tracy, TRACEE means "place of Thracius."
Girl/Female
Greek American
Reap; from Therasia.
Boy/Male
Anglo Saxon American English French
Brave.
TRACE FORMULA
TRACE FORMULA
Girl/Female
American, British, English
Special Friend
Boy/Male
Arabic, Muslim
Victory; Success
Girl/Female
Arabic, Muslim
Kitty; Kitten
Boy/Male
Indian
Cute
Boy/Male
Tamil
Prithish | பà¯à®°à¯€à®¤à®¿à®·
God of Love, Lord of the world
Girl/Female
Arabic, Greek
Virtuous; Excellent; Woman of Virtue
Male
Hindi/Indian
(रञà¥à¤œà¤¿à¤¤) Variant spelling of Hindi Ranjit, RANJEET means "victorious in battle."
Girl/Female
Greek American Biblical
Rose.
Boy/Male
Sikh
Heroic attributes
Boy/Male
Greek
Freer of horses.
TRACE FORMULA
TRACE FORMULA
TRACE FORMULA
TRACE FORMULA
TRACE FORMULA
v. t.
To trace out; to track; also, to draw out; to protact.
n.
A mark left by something that has passed along; as, the track, or wake, of a ship; the track of a meteor; the track of a sled or a wheel.
imp. & p. p.
of Trace
v.
Track; trace.
n.
A tract or area, as of land.
v.
A company of men engaged in the same occupation; thus, booksellers and publishers speak of the customs of the trade, and are collectively designated as the trade.
v. t.
To add grace notes, cadenzas, etc., to.
v. t.
To trace by scent; to track; -- a hunting term.
v. t.
To run a race with.
v. t.
To supply with heavenly grace.
v. t.
Hence, to follow the trace or track of.
v. t.
A mark left by anything passing; a track; a path; a course; a footprint; a vestige; as, the trace of a carriage or sled; the trace of a deer; a sinuous trace.
v. t.
To cause to contend in a race; to drive at high speed; as, to race horses.
v. t.
To follow the tracks or traces of; to pursue by following the marks of the feet; to trace; to trail; as, to track a deer in the snow.
v. t.
To mark out; to draw or delineate with marks; especially, to copy, as a drawing or engraving, by following the lines and marking them on a sheet superimposed, through which they appear; as, to trace a figure or an outline; a traced drawing.
n.
One who, or that which, traces.
v.
Continuity or extension of anything; as, the tract of speech.
v.
The trade winds.
n.
A mark or impression left by the foot, either of man or beast; trace; vestige; footprint.
n.
Course; way; as, the track of a comet.